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Governing Equations of Fluid Mechanics and Heat Transfer
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The integral in this equation can be evaluated if the flow is assumed barotropic. A barotropic fluid is one in which ρ is a function only of p(or a constant), that is, ρ = ρ(p). Examples of barotropic flows are as follows: Steady incompressible flow: ρ=ConstantIsentropic (constant entropy) flow (see Section 5.5.4): ρ =(Constant)p1/γ
The fluid dynamics of spin
Published in Molecular Physics, 2018
The formulation of Pauli's theory in terms of a fluid theory leads us directly to the nineteenth century work of Clebsch [47,48] and the variational formulation of fluid dynamics. Variational principles for non-magnetic barotropic fluid dynamics are well known. A four function variational formulation of Eulerian barotropic fluid dynamics was derived by Clebsch [47,48] and later by Davidov [49] whose main motivation was to quantise fluid dynamics. Since the work was written in Russian, it was unknown in the west. Lagrangian fluid dynamics (as opposed to Eulerian fluid dynamics) was formulated through a variational principle by Eckart [50]. Initial western attempts to formulate Eulerian fluid dynamics in terms of a variational principle were described by Herivel [51], Serrin [52] and Lin [53]. However, the variational principles developed by the above authors were very cumbersome containing quite a few ‘Lagrange multipliers’ and ‘potentials’. The range of the total number of independent functions in the above formulations ranges from 11 to 7 which exceeds by many of the four functions appearing in the Eulerian and continuity equations of a barotropic flow and therefore did not have any practical use or applications. Seliger and Whitham [54] have developed a variational formalism depending on only four variables for barotropic flow and thus repeated the work of Davydov's [49] which they were unaware of. Lynden-Bell and Katz [55] have described a variational principle in terms of two functions: the load λ (to be described below) and density ρ. However, their formalism contains an implicit definition for the velocity such that one is required to solve a partial differential equation in order to obtain both in terms of ρ and λ as well as its variations. Much of the same criticism holds for their general variational for non-barotropic flows [56]. Yahalom and Lynden-Bell [57] overcame this limitation by paying the price of adding an additional single function. Their formalism allowed arbitrary variations and the definition of is explicit.